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Reversibility of affine transformations
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 4 / November 2023
- Published online by Cambridge University Press:
- 08 November 2023, pp. 1217-1228
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- Article
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An element g in a group G is called reversible if g is conjugate to g−1 in G. An element g in G is strongly reversible if g is conjugate to g−1 by an involution in G. The group of affine transformations of
$\mathbb D^n$ may be identified with the semi-direct product 
$\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $, where 
$\mathbb D:=\mathbb R, \mathbb C$ or 
$ \mathbb H $. This paper classifies reversible and strongly reversible elements in the affine group 
$\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $.
